Problem 084

[jaap]

There is no need to use random number to solve this problem.

[thedoctar]

Really? Cause all the probabilities of the squares are dependent on the previous 12 squares (for six sided dice). The only way I can see to obtain the probability is to substitute heaps.

Also, do you know why I get a different answer when simulating over 3,000,000 rolls?

[brobodude]

I did something similar to thedoctor. I figured that if I simulate enough dice rolls, eventually the results would match (or be pretty darn close to) the theoretical results. Just to verify that I got similar answers each time, I wrote a program in Java to play three games each having 2,000,000 dice rolls and then compare the results. I consistently get the same answer with all three games, and even if I run the program multiple times, and the answer makes sense to me when I think about it- however, it is the wrong answer. The basic flow of each die roll is:
-Throw both die
-Move the total of both die
-Resolve rule of doubles (if necessary)
-Resolve G2J space (if necessary)
-Take CC or CH card and resolve (if necessary)
-Record ending square and repeat

Is there something really simple that I'm overlooking? Or does my approach not work? If not, why?

[thundre]

The one thing I notice that you might not be handling correctly is CH3, if you get "Go back 3 squares", that puts you on CC3, where you get another card.

Does your approach give the correct percentages for 2 6-sided dice (as given in the problem)?

Have you tested your dice model? The pair of 4-sided dice should yield p(2)=1/16, p(3)=2/16, p(4)=3/16, p(5)=4/16, p(6)=3/16, p(7)=2/16, p(8)=1/16.

[ymersvennson]

"When a player lands on CC or CH they take a card from the top of the respective pile and, after following the instructions, it is returned to the bottom of the pile."

If I draw a random card each time I land on CC or CH, I get the correct answer. But if I put them in some specific recurring order, I did not get the correct answer. I guess it is possible that there are at least some recurring orders that do not give the correct answer?

[rgraciar]

Hummm... I think this is a very interesting question, Ymersvennson.
Please, somebody of the administration team could give us an answer?
Thank you very much.

[blaise.zydeco]

In this post, I'm going to pontificate over one controversy and I'm going to raise an ambiguity in the problem statement.

Problem 84 posits a one-player game where the player moves around a Monopoly board, approximately following the rules of Monopoly. We have to assume that the problem description is complete. It doesn't ask that you research the actual rules of Monopoly. The problem has no notion of "turns" as opposed to "rolls", A "turn" is a pretty meaningless idea in a on-player game. It does not say that if you roill doubles, your "turn" is extended and you roll again. And so there is no reason to believe that a square landed on with a double should not be counted as visited.

The problem states that if you are sent to jail, you pay to get out. Suppose you are sent to jail because of the three doubles rule. You pay your way out. What happens if you now roll doubles? A case can be made from the problem statement that you go ot jail, because you have rolled three doubles in a row. But most solvers seem to assume the real Monopoly rule: if you are sent to jail and roll doubles, you do actually leave jail and are mow deemed to have rolled one previous double for the purposes of the three-double-to-jail rule.